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Computers and Chemical Engineering 31 (2007) 107–120

Application of orthogonal collocation and regression techniques forrecovering parameters of a two-pathway transdermal

drug-delivery model

Amit Goyal a, Shailaja Mandapuram a, Bozena Michniak b, Laurent Simon a,∗a Otto H. York Department of Chemical Engineering, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA

b Department of Pharmacology and Physiology, UMDNJ-New Jersey Medical School, Newark, NJ 07103, USA

Received 29 June 2005; received in revised form 30 January 2006; accepted 16 May 2006Available online 27 June 2006

bstract

This work focuses on the estimation of model parameters for a two-pathway transdermal drug-delivery system in which molecules are allowedo transfer from one pathway to the other. Orthogonal collocation, implemented in Mathematica® (Wolfram Research Inc.), was used to reducehe original system into a set of ordinary differential equations (ODEs). After selecting the optimum number of internal collocation points, aensitivity analysis was performed to determine the process parameters that significantly affected the cumulative amount of drug released. “Inilico” experiments were conducted using randomly generated values of the key input variables. A study was designed to explore the ability ofn orthogonal collocation-based optimization scheme to recover the true parameter values from simulation data. Although the solution of suchn inverse problem is non-unique, the methodology was successful at identifying the process values, in most cases. This approach avoids these of elaborate numerical inversion techniques of Laplace transforms, uses a single computational environment for generating the ODEs andstimating the parameters and does not require an analytical solution of the model. The computational approach was applied to elucidate the

ransport mechanism of caffeine through excised human skin in the presence of several permeation enhancers. The method shows that (i) the routef penetration of the drug across the skin is determined by the type of enhancers, (ii) the notable increase in steady-state transdermal flux, inomparison with control, is reflected in the estimated model parameters.

2006 Elsevier Ltd. All rights reserved.

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eywords: Orthogonal collocations; Transdermal drug delivery; Regression

. Introduction

Mathematical models, based on compartmental or diffusion-ased approaches, have been used to describe transdermal drugelivery and are indispensable tools, for allowing researcherso gain a greater insight into the mechanisms of drug action,

etabolism and transport. A clinician can use these models toevelop drug administration strategies and to select the optimumosage regimen for the treatment and prevention of disease. Tohis end, the development of accurate methods for estimating

ey model parameters is essential for medical breakthroughsnd ongoing research in controlled drug release. However, byn large, the use of mathematical tools, to estimate parameters,

∗ Corresponding author. Tel.: +1 973 596 5263; fax: +1 973 596 8436.E-mail address: laurent.simon@njit.edu (L. Simon).

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098-1354/$ – see front matter © 2006 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2006.05.007

as been restricted to compartmental models described by linearDEs or to systems that can be represented by linear partial dif-

erential equations (PDEs). This limitation is due primarily to theact that appropriate solution strategies are essential to the suc-ess of the regression method. Although analytical solutions cane obtained using Laplace transforms, PDE systems usually leado integral equations or an infinite series (Crump, 1976). In thisase, the performance of the data-fitting procedure is dependentn the number of terms included in the series approximation. Asresult, parameter estimation in PDE systems usually involvesdiscretization step, consisting of a reduction of the original

ystem into a set of ODEs and a regression step. One of thedvantages of this approach is that it is applicable to nonlinear

ystems. The main drawback is that two different software pro-rams may be necessary to accomplish the two-step process:ne for converting the original PDEs into ODEs; the other toegress the resulting ODEs.

108 A. Goyal et al. / Computers and Chemical Engineering 31 (2007) 107–120

Nomenclature

Stratum corneumCT,s drug concentration in the transcellular region

(mol/m3)CI,s drug concentration in the intercellular region

(mol/m3)DT,s drug diffusivity in the transcellular region (m2/s)DI,s drug diffusivity in the intercellular region (m2/s)

Viable epidermisCT,v drug concentration in the transcellular region

(mol/m3)CI,v drug concentration in the intercellular region

(mol/m3)DT,v drug diffusivity in the transcellular region (m2/s)DI,v drug diffusivity in the intercellular region (m2/s)

VolumesVs volume of the stratum corneum layer (m3)Vv volume of the viable epidermis layer (m3)VT,s volume of the transcellular region in the stratum

corneum (m3)VI,s volume of the intercellular region (i.e., void vol-

ume) in the stratum corneum (m3)VT,v volume of transcellular region in the viable epi-

dermis (m3)VI,v volume of the intercellular region (i.e., void vol-

ume) in the viable epidermis (m3)

Volume fractionsfs volume fraction of the intercellular region in the

stratum corneumfv volume fraction of the intercellular region in the

viable epidermis

Forward rate constants (reaction constants)Kf,s rate constant of the forward reaction in the stratum

corneum (s−1)Kf,v rate constant in the forward reaction in the viable

epidermis (s−1)

Backward rate constants (reaction constants)Kb,s rate constant of the reverse reaction in the stratum

corneum (s−1)Kb,v rate constant in the reverse reaction in the viable

epidermis (s−1)

Metabolic rate constantsKT,s rate constant of the drug in the transcellular region

of the stratum corneum (s−1)KT,v rate constant of the drug in the transcellular region

of the viable epidermis (s−1)KI,s rate constant of the drug in the intercellular region

of the in stratum corneum (s−1)KI,v rate constant of the drug in the intercellular region

of the viable epidermis (s−1)

Partition coefficientsδT drug partition coefficient from a patch (or reser-

voir) into the transcellular regionδI drug partition from a patch (or reservoir) into the

intercellular regionγT drug partition coefficient between the stratum

corneum and the viable epidermis in the transcel-lular domain

γ I drug partition coefficient between the stratumcorneum and the viable epidermis in the inter-cellular domain

FluxesJT drug permeation flux from the transcellular region

(mol/s m3)JI drug permeation flux from the intercellular region

(mol/s m3)Jtot,ss total steady-state flux (mol/s m3)Qtot cumulative amount of drug released (mol/m3)Qtot,e estimated cumulative amount of drug released

(mol/m3)

Other symbolsA effective diffusional area (m2)mdrug,s amount of drug in the stratum corneum (mol)mdrug,v amount of drug in the viable epidermis (mol)α small numerical change made in each parameter

for the sensitivity analysisp parameter vectorY vector of drug concentration values at the collo-

cation points (mol/m3)CP ratio of parameter with enhancer to control value

of parameterpar1 δIh1 (m)

YAiPiftnwvboegdwt

par2 DI,s/h21 (s−1)

Regression softwares, such as MULTI FILT (Yano,amaoka, & Tanaka, 1989), MINIM, and SCIENTIST (Roberts,nissimov, & Gonsalvez, 2001), based on numerical Laplace

nversion routines and least-square methods, are often used.DEFIT (Schittkowski, 1999), which uses FORTRAN code and

s based on methods of line, is another program that can be usedor data-fitting problems involving linear and nonlinear PDE sys-ems. However, regardless of the software used, a critical issue,ot fully addressed in parametric studies of drug transport, ishether the regression algorithm converges to the true parameteralues. Consequently, the results of these investigations have noteen instrumental in helping to elucidate potential mechanismsf drug delivery and the intrinsic effects of physical and chemicalnhancers on drug permeation. A regression analysis may sug-est that the addition of a pharmaceutical excipient enhances

rug delivery by increasing the drug diffusivity in the vehiclehen, in fact, the new formulation results in fast partitioning of

he drug into the skin. To avoid erroneous conclusions, methods

emical Engineering 31 (2007) 107–120 109

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Fig. 1. (a) Routes of drug penetration through skin. (b) Schematic representationof intercellular and transcellular volumes in stratum corneum and viable epider-mis layers VT,s: volume of the transcellular region in the stratum corneum; VI,s:volume of the intercellular region in the stratum corneum; VT,v: volume of tran-si

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A. Goyal et al. / Computers and Ch

f estimating and validating model parameters, extracted fromxperimental data, are critical for the simulation, optimizationnd design of drug-delivery patches. In the absence of such tools,t is difficult to interpret the results of “in silico” experiments.his point is not as crucial when conventional compartmentalodels are employed to describe drug absorption because of

heir empirical nature.With the widespread use of chemical enhancers (Moser,

riwet, Naik, Kalia, & Guy, 2001) and other techniques toncrease drug solubility and diffusion through the stratumorneum, parameter estimation of the original first principle-ased PDE system may be more advantageous, for design pur-oses, than compartmental models. An orthogonal collocation-ased regression approach, implemented in a single computa-ional environment, is presented in this work. The orthogonalollocation method is a well-established technique for dis-retization of PDE systems (Babu & Sastry, 1999; Finlayson,981; Simon, 2005). One of the advantages of the method ishat the collocation points are chosen automatically, which tendso decrease errors much faster than the method of line. Thispproach avoids the use of elaborate numerical inversion tech-iques of Laplace transforms, does not require an analyticalolution of the model and makes use of regression tools availablen Mathematica® (Wolfram Research Inc.). Although the exam-le presented in this work is a linear model, the method can bextended to nonlinear problems as well. Therefore, researchersan focus their attention on identifying/deriving the most rep-esentative mathematical model and developing well-designedaboratory experiments.

. Transdermal drug delivery

The development of novel approaches, such as permeationhrough the skin (Barr, 1962), for the controlled delivery ofherapeutics to their target sites has been an extensive area ofesearch. In the past, the skin was perceived as an impregnableayer limited to local administration of drugs onto its surface.n mammals, skin is the most accessible and extensive organn the body. An adult human has an average area of 1.73 m2 ofkin receiving a third of the circulating blood, at any given timePanchagnula, 1997). The main advantages of transdermal drugelivery, as compared to conventional oral administrations, are:voidance of the gastrointestinal tract, first-pass metabolism inhe liver and improved patient compliance. The permeation ofifferent chemicals through the skin is a relatively slow processnfluenced by several factors, such as solubility, diffusion effi-iency and reaction rates. These considerations have paved theay for further research concentrating on improving the perfor-ance of a rate-limiting step in the drug transport mechanism.onsequently, there is an urgent need to develop efficient meth-ds to quantify the contribution of particular pathways and tostimate key physico-chemical parameters.

The main functions of the human skin are to provide pro-

ection and sensation, control the temperature, minimize wateross, and regulate loss of salts from the body. The skin consistsf two important layers: the epidermis and the dermis (Fig. 1a).he epidermis can further be divided in two layers, namely the

slAt

cellular region in the viable epidermis; VI,v: volume of the intercellular regionn the viable epidermis.

tratum corneum and the viable epidermis or living epidermis.he principal barrier to drug permeation is the top most layer oftratum corneum, which stretches from 15 to 20 cell layers con-isting mainly of dead tissues (Blank, 1965). Both the stratumorneum and the viable epidermis consist of cells and intercel-ular spaces. The cells in the stratum corneum are long and thin,orming a dense protein matrix while the intercellular space con-ains a lipid material. Cellular and intercellular spaces in viablepidermis are aqueous in nature.

Obeying Fick’s law of diffusion, drugs are thought to perme-te through the skin, via the transcellular (through cells), inter-ellular (through spaces between cells) and transappendagealthrough hair follicles and sweat glands) pathways (Williams &arry, 1992). Mathematical models have been proposed to help

dentify suitable drugs and to estimate key parameters affectingrug transport through various skin barriers. Most of the studiesse a single-pathway approach (Kubota & Ishizaki, 1985), whichails to explain the high permeability of ionic or polar solutes,uch as glucose and tetraethylammonium bromide. The drugermeation, dominated by the lipophilic layer, is not capturedn those investigations. Berner and Cooper (1987) studied twondependent pathway models, namely polar and non-polar, tolucidate independent effects. Their two-parallel pathway modelas, however, limited to the case of steady-state drug perme-

tion.Drug permeation profiles provide insightful information on

he nature of the media and physical properties of the diffu-ant. For example, transient analysis allows the calculation of

ag times, arising from several hours to even days (Arndt’s &rndt’s, 1984; Tojo & Lee, 1989). In addition to contributing

o a more accurate description of the transport of permeants

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10 A. Goyal et al. / Computers and Ch

cross the skin, an analytical approach may drastically reducehe number of clinical trials. To this end, a representative model,s well as a suitable regression algorithm, designed to estimatehysico-chemical parameters from the permeation data, play anmportant role.

Two-pathway models were studied by Yamashita et al. (1993)ithout allowing for the interchange of species between twohases. Lee, King, and Rogers (1996) also studied a detailedwo-pathway model that included inter-conversion of the drugetween intercellular and transcellular regions. Their modeleglects the transappendageal route, as it is less relevant dueo the relatively small area covered by hair follicles and sweatlands (Lee et al., 1996). The present work focuses on the identi-cation of key parameters using the mathematical representationuggested by Lee et al. (1996). The rest of the paper is organizeds follows: (1) the two-layer transdermal model and the orthog-nal collocation-based regression approach are described; (2)aterials and methods for the controlled release experiments

re outlined; (3) simulation data are generated to illustrate thepplication of the parameter-estimation method; (4) the numer-cal tool is applied to study the effects of the type of enhancersn the transdermal delivery of caffeine; (5) results and discus-ion addressing, among other things, the use of Mathematica®

Wolfram Research Inc.) for parameter estimation and analysisre presented.

.1. Two-layer transdermal model

A two-layer model, namely the stratum corneum and viablepidermis, consisting of transcellular and intercellular phases,s considered (Fig. 1). See the work of Lee et al. (1996) for a

ore detailed description and assumptions made in deriving theodel. Some salient features of the model are described here

or completeness. Intercellular and transcellular concentrationsCI and CT, respectively) are defined as a function of time (t)nd a space variable Z, which is the depth below skin surface.he skin depth, extending from 0 to h, is further divided into

wo zones, the stratum corneum (0 to h1), designated by spaceariable x, and the viable epidermis (h1 to h2), denoted by y.ome key assumptions are made. A one-dimensional model isonstructed because the area of drug application is large com-ared to thickness of the skin. Since the time scale for loss ofead cells is large, the effects of advection and cell loss areot included. The drug is mainly transported through the cellstranscellular) or between the cells (intercellular); the transap-endageal route is not taken into consideration. Also, the effectsf drug metabolism by enzymes and drug binding in the stratumorneum are not addressed. Although some of the parameters,uch as the diffusion coefficients, may vary as functions of therug concentration (Crank, 1975), they were kept constant dur-ng the simulations.

The non-dimensionalized PDEs describing the diffusant con-

entration in the two phases and skin layers are (Lee et al., 1996):

∂C∗T,s

∂t∗= Rs

∂2C∗T,s

∂x2 + rsks(C∗I,s − C∗

T,s) − μT,sC∗T,s (1)

C

t

l Engineering 31 (2007) 107–120

∂C∗I,s

∂t∗= ∂2C∗

I,s

∂x2 + ks(C∗T,s − C∗

I,s) − μI,sC∗I,s (2)

∂C∗T,v

∂t∗= vRv

∂2C∗T,v

∂y2 + rvkv(C∗I,v − C∗

T,v) − μT,vC∗T,v (3)

∂C∗I,v

∂t∗= v

∂2C∗I,v

∂y2 + kv(C∗T,v − C∗

I,v) − μI,vC∗I,v (4)

here

= Z

h1, y = Z − h1

h2, t∗ = DI,st

h21

, C∗T,s = CT,s

δTC0,

∗I,s = CI,s

δIC0, C∗

T,v = CT,v

γTδTC0, C∗

I,v = CI,v

γIδIC0

The subscripts ‘s’ and ‘v’ denote the stratum corneum andiable epidermis and non-dimensional constants are defined assee the Nomenclature):

s = Kf,sh21

DI,s, kv = Kf,vh

21

DI,s, μT,s = KT,sh

21

DI,s,

I,s = KI,sh21

DI,s, 5μT,v = KT,vh

21

DI,s, μI,v = KI,vh

21

DI,s

nd

s = Kb,s

Kf,s, rv = Kb,v

Kf,v, Rs = DT,s

DI,s,

v = DT,v

DI,v, v = DI,vh

21

DI,sh22

The intercellular phase volume fractions in the stratumorneum and the viable epidermis are fs and fv, respectively.

The concentrations are defined as follows (Fig. 1b):

T,s = mdrug,s

VT,s

VT,s

(1 − fs)Vs= mdrug,s

(1 − fs)Vs(5)

I,s = mdrug,s

VI,s

VI,s

fsVs= mdrug,s

fsVs(6)

T,v = mdrug,v

VI,v

VI,v

(1 − fv)Vv= mdrug,v

(1 − fv)Vv(7)

I,v = mdrug,v

VI,v

VI,v

fvVv= mdrug,v

fvVv(8)

The transfer of species from one phase to another is governedy the reaction: CI ⇔ CT. To write the boundary equations, therug concentration at the skin surface assumed a constant value:0 (i.e., infinite reservoir). The partition coefficients between

he vehicle and the skin are δT and δI in the transcellular andntercellular regions, respectively. Therefore, at the skin surfacex = 0), we have

∗T,s = 1, C∗

I,s = 1 (9)

At the interface between the stratum corneum (x = 1) andhe viable epidermis (y = 0), the partition coefficients in the two

emica

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A. Goyal et al. / Computers and Ch

egions are γT and γ I:

∗T,v = C∗

T,s, C∗I,v = C∗

I,s (10)

The conditions for flux continuity at the interface betweenhe stratum corneum and viable epidermis are (Lee et al., 1996):

sRvdC∗

T,v

dy

∣∣∣∣y=0

= σrvRsdC∗

T,s

dx

∣∣∣∣x=1

(11)

dC∗I,v

dy

∣∣∣∣y=0

= σdC∗

I,s

dx

∣∣∣∣x=1

(12)

here σ = (DI,sh2)/(DI,vh1)At the end of the viable epidermis (y = 1), the drug is released

o the systemic circulation where a sink condition is used:

∗T,v = 0, C∗

I,v = 0 (13)

Assuming that the skin is initially free of drug, the followingonditions hold:

∗T,s = 0; C∗

I,s = 0; C∗T,v = 0; C∗

I,v = 0 (14)

Based on the dimensionless parameters, the fluxes are:

∗T = − dC∗

T,v

dy

∣∣∣∣y=1

= dQ∗T

dt∗(15)

nd

∗I = − dC∗

I,v

dy

∣∣∣∣y=1

= dQ∗I

dt∗(16)

here Q∗T and Q∗

I are the cumulative amount (using dimension-ess parameters) in the transcellular and intercellular regions,espectively. As a result, the dimensional cumulative amount ofrug is

tot(t) = par1

σ

(Rv

rvQ∗

T + Q∗I

)(17)

here par1 = δIh1 and t* = par2t with par2 = DI,s/h21.

. Orthogonal collocation-based regression approach

.1. Orthogonal collocation method

The system formed by Eqs. (1)–(4) can be solved by severalethods available in the literature (Beck, Cole, Haji-Sheikh, &itkouhi, 1992; Fernandes, Simon, & Loney, 2005; Mikhailov

Ozisik, 1984; Simon & Loney, 2005). Since the system isinear, a closed-from solution can be found by using Laplaceransform methods and the Residue theorem (Simon & Loney,005). This approach was not implemented in this work because

he dependence of the Eigen values on the model parameters

akes the technique inadequate for parameter estimation. Theesidue theorem is more suitable if the model parameters arenown. As discussed in Section 1, softwares, such as MULTIILT (Yano et al., 1989) can be used to numerically solve linearDEs and estimate model parameters. These programs, how-

⎢⎢⎢⎢⎢⎢⎢⎣

l Engineering 31 (2007) 107–120 111

ver, are based on Laplace-transformed solutions, which maye difficult to obtain.

The method of collocation was used (Cruz, Mendes, &agalhaes, 2002; Srivastava & Joseph, 1985; Villadsen &ichelsen, 1978; Villadsen & Sørensen, 1969; VilladsenStewart, 1967). The drug concentrations in the stratum

orneum and the viable epidermis are approximated by a finiteeries:

(z, t) ≈ a(t) + b(t) + z(1 − z)N∑

i=1

ci(t)Pi−1(z) (18)

fter scaling the thickness of both layers to 1. {Pn(z)} forms aet of polynomials such that Pn(z) is orthogonal to Pm(z) when�= n, over some interval. The locations of the collocation points

re the roots of Pn(z): z1 = 0.0, z2, . . ., zN+1, and zN+2 = 1.0.actors, such as system nonlinearities and the required accu-acy of the approximated model, play key roles in selecting theumber of collocation points (N). The permeation rate was rep-esented using seven internal collocation points. This numberriginated from a steady-state analysis of the model (see Sec-ion 7.2.1). Overall, the resulting system was reduced to a set of0 ODEs:

dC∗T,s,2

dt∗= f1(Y, p)

dC∗T,s,3

dt∗= f2(Y, p)

...dC∗

T,s,8

dt∗= f7(Y, p)

dC∗I,s,2

dt∗= f8(Y, p)

dC∗I,s,3

dt∗= f9(y, p)

...dC∗

I,s,8

dt∗= f14(Y, p)

dC∗I,v,2

dt∗= f15(Y, p)

dC∗I,v,3

dt∗= f16(Y, p)

...dC∗

T,v,8

dt∗= f21(Y, p)

dC∗I,v,2

dt∗= f22(Y, p)

dC∗I,v,3

dt∗= f23(Y, p)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

(19)

...dC∗

I,v,8

dt∗= f28(Y, p)

⎥⎥⎥⎥⎦

1 emica

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12 A. Goyal et al. / Computers and Ch

nd Eqs. (15) and (16) where

= [C∗T,s,2, C

∗T,s,3, . . . , C

∗T,s,8, C

∗I,s,2, C

∗I,s,3, . . . , C

∗I,s,8,

C∗I,v,2, C

∗I,v,3, . . . , C

∗T,v,8, C

∗I,v,2, C

∗I,v,3, . . . , C

∗I,v,8],

p = [ks, kv, μT,s, μI,s, μT,v, μI,v, rs, rv, Rs, Rv, v, σ]

.2. Nonlinear regression based on orthogonal collocationechniques

We used an approach similar to the one outlined in the pack-ge PDEFIT (Schittkowski, 1997, 1999). PDEFIT implementshe method of lines to reduce original PDE systems into ODEs,hich are then integrated using standard methods for stiff andon-stiff equations. A nonlinear least-squares problem (i.e., inhis case, minimization of the mean-squared-error between pre-icted and estimated Qtot values) is then solved using routinesuch as the Gauss–Newton-type methods. In light of the experi-nces of researchers in the use of numerical approaches to solveDEs (Babu & Sastry, 1999; Finlayson, 1981; Simon, 2005),e employed the orthogonal collocation procedure to reduce the

ransdermal drug-delivery problem in a set of ODEs. The objec-ive is to minimize the squared-error between Qtot and Qtot,e:

inp

m∑i=1

[Qtot(i) − Qtot,e(i)]2

(20)

here Qtot,e is the estimated cumulative amount of drug releasednd m is the number of data points used. A global optimizationcheme, based on Newton’s method, was implemented using the

athematica® command “NMinimize”.

. Sensitivity analysis

A sensitivity analysis was conducted to identify a set of modelarameters that significantly affected the dimensional steady-tate total flux of the drug (Jtot,ss = JI,ss + JT,ss). This approachacilitated convergence of the regression algorithm. The causalmportance of individual parameters was evaluated by slightlyhanging the elements of the parameter vector p. The remain-ng components were kept at their nominal values while theotal steady-state flux was computed. A parameter’s influencen Jtot,ss was quantified by computing the ratio dJtot,ss/dpi. Theve-point derivative rule was used to estimate dJtot,ss/dpi:

dJtot,ss

dpi

=

Jtot,ss (pi − 2�pi) − 8Jtot,ss (pi − �pi)

+ 8Jtot,ss (pi + �pi) − Jtot,ss (pi + 2�pi)

12�pi

(21)

here �pi denotes a small change in the parameter pi.

. Computer simulation experiments

The objective of the numerical study was threefold. One aimas to determine the number of internal collocation points suf-cient to assure model accuracy. Another focus was to demon-

6

n

l Engineering 31 (2007) 107–120

trate the ability of the orthogonal collocation program, writtenn Mathematica® (Wolfram Research Inc.), to reproduce resultsenerated in the work of Lee et al. (1996). This step is critical tohe methodology since the regression algorithm assumes that theumerical solution to the system is correct. The third goal waso investigate the possibility of recovering the model parametersiven a set of experimental data. These investigations are impor-ant to the pharmaceutical industry and to researchers workingn drug formulation and the elucidation of drug-delivery mech-nisms. For example, the net effect of a penetration-enhancingystem may be a 2-fold increase in the drug flux across thekin. However, a complete efficiency analysis of the transdermalrug-delivery device needs to address the pertinent pathwayshat are responsible for the enhancement. This engineering-ased approach would allow drug developers to focus on ando regulate the rate-limiting steps. The alternative is to imple-

ent a prohibitively expensive trial-and-error procedure, whichonsists of conducting numerous laboratory experiments andomparing the results. The focus of this work is to assess whetherhe regression routine, described in Section 3.2, was able toetect subtle changes in some pathway parameters solely basedn the cumulative amount of drug released.

The nominal model parameters are (Lee et al., 1996):T,s = 10−16 m2/s, DI,s = 10−14 m2/s, DT,v = 10−12 m2/s, DI,v =0−12 m2/s, KT,s = 10−10 s−1, KI,s = 10−10 s−1, KT,v = 10−4 s−1,I,v = 10−7 s−1, Kf,s = 10−4 s−1, Kb,s = 10−3 s−1, Kf,v =0−2 s−1, Kb,v = 10−3 s−1, δT = 0.01, δI = 0.1, h1 = 20 �m,2 = 130 �m. As a result, the non-dimensional constants are:s = 4, kv = 400, μT,s = 4 × 10−6, μI,s = 4 × 10−6, μT,v = 4,I,v = 4 × 10−3, rs = 10, rv = 0.1, Rs = 10−2, Rv = 1, v = 2.367,= 0.65. The lengths of the stratum corneum (h1) and the

iable epidermis (h2) have been estimated in previous studies.ee et al. (1996) provided the physical significance of eacharameter and outlined methodologies to estimate their values.xtracted skin lipids were used to experimentally determine

he intercellular diffusivity in the stratum corneum (DI,s)Lange-Lieckfeldt & Lee, 1992; Lee et al., 1996); the approachook into account the path tortuosity (Flynn, 1989; Siegel,989). The small value of DT,s is due, in part, to the denserotein matrix formed by the cells (Lee et al., 1996). Drugiffusivities, in the intercellular region of the viable epidermisDI,v and DT,v), were calculated on the basis of a hinderedqueous environment (Scheuplein, 1967). Based on the analysisy Lee et al., ks and kv indicate interconversion rates; μT,s,I,s, μT,v, μI,v denote metabolic rates; rs and rv are ratios of

ntercellular and transcellular drug concentrations; Rs and Rvompare diffusivity coefficients; v indicates the main barrier tontercellular transport (i.e., the viable epidermis or the stratumorneum).

. Controlled release of 1,3,7-trimethylxanthine in theresence of enhancers

.1. Materials

The permeation of hydrophilic caffeine (1,3,7-trimethylxa-thine) through excised dermatomed (500 �m) cadaver human

A. Goyal et al. / Computers and Chemical Engineering 31 (2007) 107–120 113

Table 1Steady-state flux analysis to determine the optimum number of collocation points

Number of internalcollocation points, N

Total number ofcollocation points,N1 = N + 2

Steady-statetranscellular flux, JT,ss

Steady-stateintercellular flux, JI,ss

Steady-state totalflux, Jtot,ss

Difference in twoconsecutive totalsteady-state flux values,Jtot,ss(N + 1) − Jtot,ss(N)

1 3 3.70E−11 −1.16E−11 2.54E−11 N/A3 5 2.60E−11 7.49E−13 2.67E−11 1.33E−125 7 2.44E−11 2.34E−12 2.67E−11 2.56E−147 9 2.43E−11 2.45E−12 2.67E−11 2.70E−159 11 2.43E−11 2.45E−12 2.67E−11 5.36E−16

1 2

T ate flu

sPrctd2&mtpcmoetcds

6

fiahr03mioccwrwwpacr

7

7

7

tsEMiSrl2

7

lware shown in Figs. 2–4. Figs. 2 and 3 represent the normalizedtranscellular and intercellular drug concentrations, respectively(compare with Fig. 2a and b, Lee et al., 1996). The cumulativeamount of drug released is shown in Fig. 4 (compare with Fig. 5,

1 13 2.43E−11

he analysis is based on the difference between two consecutive total steady-st

kins (National Disease Research Interchange, Philadelphia,A) was investigated in the presence of the following generallyecognized as safe (GRAS) enhancers: oleic acid, menthone,ineole, terpineol and azone. Although skin pretreatments withhese enhancers have been found to increase the transport ofrugs though the skin (El-Kattan, Asbill, Kim, & Michniak,001; El-Kattan, Asbill, & Michniak, 2000; Yamune, Williams,

Barry, 1995), no support, based on the relative importance ofultiple-pathway model parameters, was provided to explain

he laboratory data. To prepare the saturated drug solution,ropylene glycol was added to a specific amount of caffeineontained in a test tube. The resulting solution was thoroughlyixed using a vortex mixer and maintained at a temperature

f 32 ± 0.5 ◦C (surface temperature of skin) for 2 days (mixedvery 2–3 h) prior to each experiment. Phosphate buffer solu-ion (PBS), to be used in 5.1-ml vertical glass Franz diffusionells (Permegear Inc., Bethlehem, PA), was prepared byissolving PBS tablets in distilled water (one tablet per 100 mlolution).

.2. Experimental procedure

The receiver compartment of the Franz diffusion cell waslled with the buffer solution, stirred at 600 rpm and maintainedt 37 ± 0.5 ◦C throughout the experiments. Small pieces ofuman skin were cut and clamped between the donor andeceiver compartments with an exposed diffusion area of.64 cm2. The skin membranes were allowed to hydrate for0 min to allow equilibration of the membrane. One hundredicroliter of the drug mixed with an enhancer was placed

n donor chambers. Since the enhancers contain essentialils, which are volatile at room temperature, the cells wereovered with a ParafilmTM. No enhancer was present in theontrolled experiments. Three hundred microliter samplesere collected from the receiver compartment and immediately

eplaced with an equal amount of fresh buffer solution. Samplesere taken at predetermined times over 24 h. All samples

ere analyzed for drug content using an Agilent 1100 highressure liquid chromatograph (HPLC) equipped with a dioderray and a C-18 reverse-phase column. The mobile phaseomposition was 20:70:10 of methanol, water and acetonitrile,espectively.

Fs

.45E−12 2.67E−11 2.04E−16

x values.

. Results and discussions

.1. Computer simulation studies

.1.1. Number of collocation pointsThe total number of collocation points (N + 2), to be used in

he transient problem, was determined by solving the steady-tate and dimensional version of the model represented byqs. (1)–(4) and (9)–(14). The procedure, implemented inathematica® (Wolfram Research Inc.), consisted of increas-

ng N while monitoring the difference Jtot,ss(N + 1) − Jtot,ss(N).even internal collocation points (N = 7) were sufficient toepresent the model (Table 1). Normalized quantities areisted to allow comparison with the steady-state flux of.47 × 10−11 mol/s m3 reported by Lee et al. (1996).

.1.2. Solution of the transient modelThe transient model was solved using seven internal col-

ocation points to ascertain the performance of the algorithmritten in Mathematica® (Wolfram Research Inc.). The results

ig. 2. Transient model solution with seven internal collocation points: tran-cellular concentrations as a function of depth at different time values.

114 A. Goyal et al. / Computers and Chemical Engineering 31 (2007) 107–120

Fc

Ltififdtwltdiepp

7d

uTSt

Fl

Table 2Sensitivity analysis to estimate the relative influence of a parameter pi on thesteady-state flux Jtot,ss

Parameter α

0.1 0.01 0.001 0.0001

DT,s 3247.60 3247.60 3247.60 3247.60DI,s 1190.25 1190.25 1190.25 1190.25DT,v 2.34E−04 2.34E−04 2.34E−04 2.34E−04DI,v 50.30 50.30 50.30 50.30KT,s 0.00 0.00 0.00 0.00KI,s 0.00 0.00 0.00 0.00KT,v −4.64E−17 −4.64E−17 −4.64E−17 −4.64E−17KI,v −9.77E−15 −9.65E−15 −8.20E−15 −1.40E−15Kf,s 3.25E−09 3.25E−09 3.25E−09 3.25E−09Kb,s −3.25E−10 −3.25E−10 −3.25E−10 −3.25E−10Kf,v −3.25E−11 −3.25E−11 −3.25E−11 −3.25E−11Kb,v 3.25E−10 3.25E−10 3.25E−10 3.25E−10γT 3.25E−14 3.25E−14 3.25E−14 3.25E−14γ I 5.00E−10 5.00E−10 5.00E−10 5.00E−10δT 3.25E−11 3.25E−11 3.25E−11 3.25E−11δI 6.22E−10 6.22E−10 6.22E−10 6.22E−10

Ac

r0docPeootrtei

ig. 3. Transient model solution with seven internal collocation points: inter-ellular concentrations as a function of depth at different time values.

ee et al., 1996). Fig. 2 shows that the drug molecules, using theranscellular pathway, penetrated through the stratum corneumn less than 3 h—the concentration did not change significantlyrom 2.4 to 24 h. The transport rate was considerably reducedn the viable epidermis (Fig. 2). The sharp increase at the inter-ace between the stratum corneum and the viable epidermis isue to the assumption of rapid local equilibrium, although therend is more pronounced in Fig. 2 than in Fig. 3. Note that CI,sas one order-of-magnitude higher than CT,s while CI,v remains

ower than CT,v in the course of the simulation. The model solu-ion is instructive in that one can pinpoint the location of therug molecules at any time. More importantly, “in silico” exper-ments can help anticipate the speed at which toxic chemicalsnter the blood stream and cause irreversible damage. For controlurposes, drug molecules can be induced to travel a particularathway by using appropriate chemical or physical enhancers.

.1.3. Model parameters from the cumulative amount ofrug-released data

The relative influence of pi on the steady-state flux was eval-

ated in order to reduce the dimension of the parameter vector.o this end, a sensitivity analysis was performed as described inection 4. While making small changes in pi of size �Pi = αpi,

he steady-state flux was monitored (Table 2). To verify the accu-

ig. 4. Cumulative amount of drug released as a function of time for the simu-ation study.

tl1d(cwa

71

ammrσ

t

fter making a small change in pi of size �pi = αpi, the ratio dJtot,ss/dpi isalculated using the five-point derivative rule.

acy of the five-point derivative rule, α was varied from 0.1 to.0001. The most significant effect was due to changes in theiffusion coefficients: DT,s, DI,s, DI,v and DT,v (in decreasingrder of relevance). It is preferable to develop a mathemati-al framework in which all of the parameters can be recovered.reliminary results (not shown) revealed that the mean-squared-rror between Qtot and Qtot,e remained constant during iterationf some of the parameters. This observation promotes the usef an input sensitivity analysis to assess the relative contribu-ion of each parameter. A small number of factors were largelyesponsible for the apparent change in the minimum value ofhe objective function. Consequently, it is more appropriate tostimate the remaining parameters using correlations that takento account properties of the drugs and of the skin.

The diffusion coefficients were estimated with an optimiza-ion routine in Mathematica®: “NMinimize” subject to the fol-owing constraints 10−17 < DT,s < 10−15, 10−15 < DI,s < 10−13,0−13 < DT,v < 10−11, 10−13DI,v < 10−11. The iteration proce-ure was repeated five times using randomized starting valuesTable 3). Overall, the estimated diffusion coefficients are inlose agreement with the reference values, although convergenceas not achieved, in a few cases. The values of Rs, Rv, v and σ

re also listed.

.2. The effects of enhancers on the transdermal release of,3,7-trimethylxanthine

To apply the findings of the simulation study to the perme-tion of caffeine across the skin, we used interconversion andetabolic rate constants given in Lee et al. (1996), and esti-

ated dimensionless parameters that took into consideration the

elative importance of diffusion coefficients (i.e., Rs, Rv, v, and). Since these numbers are expected to change with the pene-

rant/formulation, the methodology only computed approximate

A. Goyal et al. / Computers and Chemical Engineering 31 (2007) 107–120 115

Table 3The model parameters estimated using an optimization routine in Mathematica®

Error DI,s DI,v DT,s DT,v Rs Rv v σ

0.00E+00 1.00E−02 1.00E+00 1.00E−04 1.00E+00 1.00E−02 1.00E+00 2.37E+00 6.50E−011.01E−10 9.93E−03 1.02E+00 6.34E−04 9.99E−01 6.38E−02 9.79E−01 2.43E+00 6.33E−014.94E−08 9.49E−03 8.73E+00 6.09E−04 1.66E−01 6.42E−02 1.90E−02 2.18E+01 7.07E−026.97E−07 1.12E−02 1.56E−01 1.72E−04 1.06E+00 1.54E−02 6.79E+00 3.30E−01 4.67E+005.69E−05 9.27E−03 9.99E+00 9.39E−04 1.00E−01 1.01E−01 1.00E−02 2.55E+01 6.03E−021.29E−02 7.16E−03 4.38E−01 6.90E−04 1.00E+01 9.64E−02 2.28E+01 1.45E+00 1.06E+00

0.00E+00 7.00E−02 3.33E−01 2.00E−05 9.00E+00 2.86E−04 2.70E+01 1.13E−01 1.37E+012.12E−05 6.23E−02 4.86E−01 1.00E−05 1.00E+01 1.61E−04 2.06E+01 1.85E−01 8.33E+002.63E−05 6.12E−02 5.14E−01 5.21E−04 9.99E+00 8.51E−03 1.94E+01 1.99E−01 7.74E+001.80E−03 4.99E−02 2.15E+00 7.62E−04 9.27E+00 1.53E−02 4.31E+00 1.02E+00 1.51E+002.26E−03 4.47E−02 1.00E+01 1.00E−03 1.00E+01 2.24E−02 1.00E+00 5.29E+00 2.91E−012.27E−03 4.48E−02 1.00E+01 1.71E−04 1.00E+01 3.82E−03 1.00E+00 5.28E+00 2.91E−01

0.00E+00 1.43E−03 3.00E+00 5.00E−04 1.11E−01 3.50E−01 3.70E−02 4.97E+01 3.10E−022.22E−14 1.41E−03 3.08E+00 7.48E−04 1.02E−01 5.30E−01 3.31E−02 5.17E+01 2.98E−022.12E−12 1.48E−03 3.06E+00 1.19E−05 1.04E−01 8.04E−03 3.40E−02 4.89E+01 3.14E−022.70E−11 1.48E−03 3.10E+00 1.00E−05 1.00E−01 6.76E−03 3.23E−02 4.96E+01 3.10E−021.38E−10 1.43E−03 1.44E−01 9.54E−04 4.18E−01 6.67E−01 2.90E+00 2.38E+00 6.45E−012.09E−03 1.00E−03 1.00E+01 1.00E−05 1.00E−01 1.00E−02 1.00E−02 2.37E+02 6.50E−03

0.00E+00 3.33E−03 1.43E−01 9.00E−04 5.00E+00 2.70E−01 3.50E+01 1.02E+00 1.51E+002.82E−10 3.41E−03 1.83E−01 1.66E−05 5.01E+00 4.87E−03 2.74E+01 1.27E+00 1.21E+001.60E−08 3.16E−03 9.92E+00 7.38E−04 4.15E+00 2.34E−01 4.18E−01 7.43E+01 2.07E−021.78E−08 3.19E−03 9.18E+00 4.59E−04 4.23E+00 1.44E−01 4.61E−01 6.81E+01 2.26E−021.85E−08 3.30E−03 1.37E+00 1.94E−05 5.03E+00 5.88E−03 3.67E+00 9.83E+00 1.57E−013.79E−05 3.18E−03 3.35E−01 5.02E−04 9.64E+00 1.58E−01 2.88E+01 2.49E+00 6.17E−01

0.00E+00 5.00E−02 9.00E+00 7.00E−04 3.33E−01 1.40E−02 3.70E−02 4.26E+00 3.61E−019.59E−10 5.01E−02 8.05E+00 1.00E−03 4.40E−01 2.00E−02 5.47E−02 3.80E+00 4.05E−011.17E−08 5.05E−02 6.83E+00 6.79E−04 5.74E−01 1.34E−02 8.40E−02 3.20E+00 4.81E−013.40E−08 5.16E−02 5.18E+00 8.00E−04 7.51E−01 1.55E−02 1.45E−01 2.38E+00 6.47E−011.43E−06 5.62E−02 2.42E+00 6.83E−04 1.03E+00 1.22E−02 4.26E−01 1.02E+00 1.51E+001.03E−05 6.02E−02 1.55E+00 2.12E−05 1.11E+00 3.52E−04 7.16E−01 6.09E−01 2.52E+00

0.00E+00 1.11E−03 3.00E+00 1.43E−05 5.00E+00 1.29E−02 1.67E+00 6.40E+01 2.41E−026.53E−12 1.10E−03 3.19E+00 7.59E−05 4.98E+00 6.90E−02 1.56E+00 6.86E+01 2.24E−021.47E−11 1.05E−03 1.82E+00 6.63E−04 5.12E+00 6.31E−01 2.81E+00 4.10E+01 3.75E−021.09E−10 1.07E−03 2.31E+00 4.36E−04 5.06E+00 4.07E−01 2.19E+00 5.11E+01 3.01E−023.35E−10 1.10E−03 1.00E+01 1.07E−05 4.22E+00 9.73E−03 4.22E−01 2.15E+02 7.15E−035.63E−10 1.07E−03 9.98E−01 5.24E−04 5.18E+00 4.90E−01 5.19E+00 2.21E+01 6.97E−02

0.00E+00 5.00E−02 7.00E+00 3.00E−04 1.11E−01 6.00E−03 1.59E−02 3.31E+00 4.64E−012.03E−10 1.97E−03 6.99E+00 6.46E−04 1.11E−01 3.28E−01 1.59E−02 8.40E+01 1.83E−026.93E−10 2.08E−03 3.02E−01 1.00E−05 8.38E−01 4.81E−03 2.77E+00 3.44E+00 4.48E−012.49E−09 2.02E−03 1.06E−01 1.00E−03 1.00E−03 4.95E−01 9.43E−03 1.24E+00 1.24E+001.68E−05 1.89E−03 8.57E+00 1.29E−04 1.00E−01 6.83E−02 1.17E−02 1.07E+02 1.43E−026.86E−04 1.30E−03 9.61E+00 5.74E−04 7.11E+00 4.42E−01 7.40E−01 1.75E+02 8.79E−03

0.00E+00 9.00E−02 7.00E+00 3.33E−05 2.00E−01 3.70E−04 2.86E−02 1.84E+00 8.36E−011.19E−08 9.12E−02 5.92E+00 4.98E−04 3.20E−01 5.46E−03 5.41E−02 1.54E+00 1.00E+002.26E−08 8.96E−02 7.76E+00 4.93E−04 1.14E−01 5.50E−03 1.47E−02 2.05E+00 7.51E−013.07E−08 8.96E−02 7.86E+00 1.00E−03 1.03E−01 1.12E−02 1.31E−02 2.08E+00 7.41E−011.46E−07 9.56E−02 4.22E+00 −4.00E−04 5.02E−01 4.18E−03 1.19E−01 1.04E+00 1.47E+006.19E−07 9.85E−02 3.57E+00 9.99E−04 5.70E−01 1.01E−02 1.60E−01 8.58E−01 1.79E+00

0.00E+00 2.00E−03 3.00E+00 1.11E−05 7.00E+00 5.55E−03 2.33E+00 3.55E+01 4.33E−023.15E−09 1.99E−03 1.72E−01 5.65E−04 7.11E+00 2.84E−01 4.13E+01 2.05E+00 7.52E−017.64E−09 2.07E−03 1.03E−01 5.77E−05 6.97E+00 2.79E−02 6.77E+01 1.18E+00 1.31E+002.22E−07 1.93E−03 8.29E+00 4.92E−04 7.16E+00 2.55E−01 8.64E−01 1.02E+02 1.51E−025.52E−07 1.97E−03 1.34E+00 2.35E−04 8.28E+00 1.19E−01 6.18E+00 1.61E+01 9.56E−021.29E−06 1.92E−03 9.60E+00 4.63E−04 8.04E+00 2.41E−01 8.38E−01 1.18E+02 1.30E−02

0.00E+00 3.33E−03 5.00E+00 1.11E−05 7.00E+00 3.33E−03 1.40E+00 3.55E+01 4.33E−025.88E−10 3.28E−03 8.80E+00 3.17E−04 6.56E+00 9.66E−02 7.45E−01 6.35E+01 2.42E−028.42E−10 3.25E−03 4.23E+00 9.22E−04 7.08E+00 2.84E−01 1.67E+00 3.08E+01 4.99E−026.08E−07 3.57E−03 1.00E−01 8.80E−05 6.49E+00 2.46E−02 6.49E+01 6.63E−01 2.32E+002.45E−06 3.22E−03 8.64E+00 7.47E−04 8.05E+00 2.32E−01 9.32E−01 6.35E+01 2.42E−02

116 A. Goyal et al. / Computers and Chemical Engineering 31 (2007) 107–120

Table 3 (Continued )

Error DI,s DI,v DT,s DT,v Rs Rv v σ

6.05E−06 3.34E−03 3.43E−01 4.50E−04 9.67E+00 1.35E−01 2.82E+01 2.43E+00 6.33E−01

0.00E+00 5.00E−02 9.00E+00 3.33E−05 1.43E−01 6.66E−04 1.59E−02 4.26E+00 3.61E−012.07E−10 5.00E−02 9.24E+00 8.80E−05 1.16E−01 1.76E−03 1.26E−02 4.37E+00 3.52E−015.44E−10 5.06E−02 5.77E+00 7.41E−04 5.04E−01 1.46E−02 8.73E−02 2.70E+00 5.70E−015.68E−10 5.05E−02 5.86E+00 9.92E−04 4.95E−01 1.96E−02 8.45E−02 2.75E+00 5.60E−011.12E−09 5.01E−02 7.03E+00 1.94E−04 3.65E−01 3.87E−03 5.19E−02 3.32E+00 4.63E−013.36E−05 6.55E−02 9.46E−01 1.01E−05 9.79E−01 1.54E−04 1.03E+00 3.42E−01 4.50E+00

0.00E+00 1.43E−03 3.33E−01 9.00E−04 2.00E−01 6.29E−01 6.01E−01 5.51E+00 2.79E−012.84E−12 1.47E−03 1.23E+00 1.64E−04 1.04E−01 1.12E−01 8.46E−02 1.98E+01 7.77E−021.48E−11 1.45E−03 1.12E−01 9.92E−04 2.22E−01 6.84E−01 1.98E+00 1.83E+00 8.42E−019.37E−11 1.49E−03 1.27E+00 1.00E−05 1.00E−01 6.71E−03 7.87E−02 2.02E+01 7.63E−021.14E−10 1.48E−03 1.26E+00 8.97E−05 1.00E−01 6.06E−02 7.94E−02 2.02E+01 7.63E−023.69E−02 1.00E−03 1.00E+01 1.00E−05 1.00E−01 1.00E−02 1.00E−02 2.37E+02 6.50E−03

0.00E+00 7.00E−02 2.00E−01 3.00E−04 1.11E−01 4.29E−03 5.55E−01 6.76E−02 2.28E+017.47E−08 5.41E−02 3.22E−01 9.99E−04 1.00E−01 1.85E−02 3.11E−01 1.41E−01 1.09E+018.07E−08 5.39E−02 3.23E−01 9.96E−04 1.00E−01 1.85E−02 3.10E−01 1.42E−01 1.08E+015.29E−05 1.00E−01 1.00E−01 1.74E−05 1.27E−01 1.74E−04 1.27E+00 2.37E−02 6.50E+015.34E−02 3.00E−03 9.02E+00 7.82E−04 7.92E+00 2.61E−01 8.78E−01 7.12E+01 2.16E−025.39E−02 3.11E−03 3.36E−01 5.12E−04 9.65E+00 1.65E−01 2.87E+01 2.56E+00 6.02E−01

0.00E+00 5.00E−02 3.00E+00 1.43E−05 1.11E−01 2.86E−04 3.70E−02 1.42E+00 1.08E+006.79E−10 5.01E−02 2.92E+00 3.34E−05 1.20E−01 6.67E−04 4.11E−02 1.38E+00 1.12E+007.53E−10 4.99E−02 3.09E+00 1.00E−05 1.01E−01 2.00E−04 3.27E−02 1.47E+00 1.05E+009.25E−10 4.99E−02 3.10E+00 6.97E−05 1.00E−01 1.40E−03 3.23E−02 1.47E+00 1.05E+002.20E−06 5.75E−02 1.14E+00 8.85E−04 3.09E−01 1.54E−02 2.71E−01 4.69E−01 3.28E+002.04E−01 2.41E−02 1.00E+01 9.00E−04 1.00E−01 3.73E−02 1.00E−02 9.82E+00 1.57E−01

0.00E+00 9.00E−02 3.00E+00 1.43E−05 5.00E+00 1.59E−04 1.67E+00 7.89E−01 1.95E+007.32E−09 8.96E−02 3.03E+00 7.90E−04 5.00E+00 8.82E−03 1.65E+00 8.00E−01 1.92E+002.41E−08 8.98E−02 3.03E+00 1.97E−05 5.00E+00 2.19E−04 1.65E+00 7.99E−01 1.93E+005.62E−08 8.95E−02 3.07E+00 7.46E−04 5.00E+00 8.34E−03 1.63E+00 8.12E−01 1.89E+003.13E−05 8.38E−02 4.92E+00 2.19E−05 5.05E+00 2.61E−04 1.03E+00 1.39E+00 1.11E+006.17E−05 7.78E−02 9.80E+00 2.45E−04 4.73E+00 3.15E−03 4.83E−01 2.98E+00 5.16E−01

0.00E+00 3.33E−03 1.43E−01 5.00E−04 9.00E+00 1.50E−01 6.29E+01 1.02E+00 1.51E+001.96E−09 3.31E−03 2.50E−01 2.76E−04 9.19E+00 8.34E−02 3.68E+01 1.79E+00 8.61E−012.34E−08 3.21E−03 9.96E+00 1.27E−05 8.55E+00 3.96E−03 8.58E−01 7.34E+01 2.09E−022.37E−08 3.21E−03 1.00E+01 1.00E−05 8.59E+00 3.12E−03 8.59E−01 7.37E+01 2.09E−028.79E−08 3.26E−03 3.20E−01 5.33E−04 9.65E+00 1.63E−01 3.02E+01 2.32E+00 6.62E−012.86E−07 3.15E−03 8.57E+00 7.04E−04 8.11E+00 2.23E−01 9.46E−01 6.44E+01 2.39E−02

0.00E+00 9.00E−02 2.00E−01 7.00E−04 3.00E+00 7.78E−03 1.50E+01 5.26E−02 2.93E+011.15E−04 6.31E−02 4.80E−01 1.35E−04 3.27E+00 2.14E−03 6.81E+00 1.80E−01 8.54E+001.40E−04 6.13E−02 5.31E−01 7.07E−04 3.26E+00 1.15E−02 6.14E+00 2.05E−01 7.50E+005.81E−04 4.62E−02 4.05E+00 1.73E−04 3.35E+00 3.74E−03 8.27E−01 2.07E+00 7.41E−016.15E−04 4.38E−02 8.50E+00 9.93E−04 2.94E+00 2.27E−02 3.46E−01 4.59E+00 3.35E−016.20E−04 4.48E−02 6.19E+00 4.43E−04 3.18E+00 9.89E−03 5.14E−01 3.27E+00 4.70E−01

0.00E+00 7.00E−02 3.33E−01 1.11E−05 2.00E−01 1.59E−04 6.01E−01 1.13E−01 1.37E+013.09E−06 4.74E−02 1.30E+00 1.00E−05 1.02E−01 2.11E−04 7.85E−02 6.49E−01 2.37E+003.13E−06 4.73E−02 1.32E+00 1.00E−05 1.00E−01 2.11E−04 7.58E−02 6.61E−01 2.33E+003.14E−06 4.72E−02 1.32E+00 5.59E−04 1.00E−01 1.18E−02 7.58E−02 6.62E−01 2.32E+003.14E−06 4.72E−02 1.32E+00 4.25E−04 1.00E−01 9.00E−03 7.58E−02 6.62E−01 2.32E+002.59E−03 1.00E−01 1.00E−01 1.00E−03 2.56E−01 1.00E−02 2.56E+00 2.37E−02 6.50E+01

0.00E+00 3.00E−02 1.43E−01 9.00E−04 5.00E+00 3.00E−02 3.50E+01 1.13E−01 1.36E+014.52E−05 2.23E−02 1.90E+00 1.00E−05 6.02E+00 4.48E−04 3.17E+00 2.02E+00 7.63E−016.08E−05 2.09E−02 9.69E+00 6.34E−04 5.52E+00 3.03E−02 5.70E−01 1.10E+01 1.40E−016.21E−05 2.10E−02 7.16E+00 9.85E−04 5.79E+00 4.69E−02 8.09E−01 8.07E+00 1.91E−011.21E−04 2.10E−02 9.99E+00 8.82E−04 4.87E+00 4.20E−02 4.87E−01 1.13E+01 1.37E−013.39E−04 2.43E−02 6.43E−01 7.71E−04 4.72E+00 3.17E−02 7.34E+00 6.26E−01 2.46E + 00

0.00E+00 3.33E−03 9.00E+00 7.00E−04 2.00E−01 2.10E−01 2.22E−02 6.40E+01 2.41E−023.17E−11 3.38E−03 2.86E+00 1.76E−04 8.79E−01 5.21E−02 3.07E−01 2.00E+01 7.68E−027.06E−11 3.37E−03 2.63E+00 3.92E−04 9.03EV01 1.16E−01 3.43E−01 1.85E+01 8.33E−022.04E−10 3.36E−03 2.64E+00 4.69E−04 9.03E−01 1.40E−01 3.42E−01 1.86E+01 8.27E−021.19E−07 3.39E−03 9.94E+00 1.74E−04 1.00E−01 5.13E−02 1.01E−02 6.94E+01 2.22E−02

A. Goyal et al. / Computers and Chemical Engineering 31 (2007) 107–120 117

Table 3 (Continued )

Error DI,s DI,v DT,s DT,v Rs Rv v σ

1.37E−03 2.53E−03 1.29E+00 4.63E−04 8.53E+00 1.83E−01 6.61E + 00 1.21E+01 1.27E−01

0.00E+00 2.00E−03 3.00E+00 1.11E−05 1.43E−01 5.55E−03 4.77E−02 3.55E+01 4.33E−022.53E−12 1.92E−03 1.27E+00 9.67E−04 3.33E−01 5.04E−01 2.62E−01 1.57E+01 9.83E−026.08E−12 1.92E−03 3.37E+00 9.99E−04 1.00E−01 5.20E−01 2.97E−02 4.15E+01 3.70E−021.38E−11 1.95E−03 3.32E+00 6.52E−04 1.06E−01 3.34E−01 3.19E−02 4.03E+01 3.82E−021.34E−09 1.98E−03 1.10E+00 3.43E−04 3.51E−01 1.73E−01 3.19E−01 1.31E+01 1.17E−017 3E+0

T ts. Thc emain

sto

tttknpppmotTuaaal

vett

wT(w1oerbcbteu

neatawa

TE

CC

OC

CC

MC

TC

AC

Cc

.79E−04 1.01E−03 1.00E+01 7.67E−05 1.9

he iteration procedure was repeated five times using randomized starting poinoefficients represent the best solution in a least-mean-square-error sense; the r

olutions. However, the approach might help to identify poten-ial pathways and key factors governing transdermal transportf caffeine.

Eq. (17) requires that par1 (δIh1) and par2 (DI,s/h21) be added

o the parameter spaces. In fact, preliminary studies show thathe algorithm could not converge to values of Rs, Rv, v, and σ

hat accurately fit the experimental data when par1 and par2 areept at their nominal values. This may be due to the hydrophilicature of caffeine; δI is a measure of lipophilicity. Results of thearametric study are listed in Table 4. It took several startingoints (>5) to obtain adequate match between experimental andredicted Q profiles. The results that corresponded to the mini-um squared-error are reported. The control (con) and the type

f enhancer (en) used are given in the first column, along withhe steady-state flux (Jss,en or Jss,con) and the ratio Jss,en/Jss,con.he CP value is the ratio of a parameter when an enhancer issed to the parameter value in the control. Oleic acid provided36.7-fold increase in caffeine flux followed by azone, with35.6-fold increase. Cineole showed only mild enhancement

ctivity with a 11.5-fold increase. Menthone and terpineol wereess effective.

The predicted drug profiles matched the experimental data

ery well (Fig. 5). Based on the CP values (Table 4), the fluxnhancements, observed at steady-state, were the results ofhe contributions of individual transport pathways. For men-hone, terpineole, azone and the control, the stratum corneum

rlom

able 4stimated parameter values for the permeation of caffeine in skin

Error Rs

ontrol (5.354E−01; 1.000E+00) 1.19E−12 9.81E−P 1.00E+0

leic acid (1.966E+01; 3.672E+01) 1.36E−10 1.00E+0P 1.02E+0

ineole (6.152E+00; 1.149E+01) 1.65E−10 1.00E−P 1.02E−enthone (1.975E+00; 3.688E+00) 3.09E−13 1.00E+0P 1.02E+0

erpineole (1.630E+00; 3.044E+00) 3.44E−12 9.09E−P 9.27E−zone (1.908E+01; 3.56E+01) 1.72E−11 9.88E−P 1.01E+0

hange in cumulative drug and parameters due to enhancers. The control, enhancerolumn. The CP value is the ratio of a parameter when an enhancer is used to the par

0 7.59E−02 1.93E−01 2.34E+02 6.57E−03

e values in bold represent the parameters to be estimated; the italics diffusioning four solutions are sorted in decreasing order of the error.

as the main barrier to intercellular drug transport (v � 1).he high CP for the parameter Rv when oleic acid was used

1.00E+04) may explain the significant flux increase. Azone,hich was an equally efficient enhancer, has a CP value of.43. However, the relatively low CP value for v in the casef oleic acid, as compared to azone, may help to explain thexperimental results. Also, based on the regression analysis, theatio par1,oleic acid/par1,azone is 7.5E−02; the partition coefficientetween the donor and the intercellular phase in the stratumorneum is less when oleic acid is used. Similar analysis cane done on all the enhancers to explain the change in Jss fromhe base line. However, these conclusions need to be verified byxperiment since the solution to the optimization problem is notnique.

Several key findings emerge from this contribution. The sig-ificant parameters of a transdermal drug-delivery model werestimated using orthogonal collocation methods and regressionlgorithms in a single computing environment. A method to par-ially elucidate the mechanism of drug delivery through the skinnd quantify the overall contribution of penetration enhancersas investigated. The findings could be used to design precise

nd efficacious drug-release devices and meet specific end-use

equirements. The predictive capability of the methodology, out-ined in this work, allows for the assessment of the performancef drug-delivery devices without conducting numerous experi-ents. This effort is aimed at improving the transdermal delivery

Rv v σ

01 1.00E−02 5.13E+01 6.15E+010 1.00E+00 1.00E+00 1.00E+00

0 1.00E+02 1.00E−02 1.00E+020 1.00E+04 1.95E−04 1.63E+00

04 8.76E+01 5.62E−01 2.69E+0004 8.75E+03 1.10E−02 4.37E−02

0 2.93E−02 1.00E+02 1.00E+020 2.93E+00 1.95E+00 1.63E+00

01 1.34E−02 3.41E+01 1.00E+0201 1.34E+00 6.65E−01 1.63E+00

01 1.43E−02 6.54E+01 4.25E+010 1.43E+00 1.27E+00 6.90E−01

i, steady-state flux and ratio Jss,i/Jss,con (Jss,i; Jss,i/Jss,con) are listed in the firstameter value in the control.

118 A. Goyal et al. / Computers and Chemical Engineering 31 (2007) 107–120

Fig. 5. Cumulative amount of drug released as a function of time: experimental and estimated data. The subplots are (a) control, (b) oleic acid, (c) cineole, (d)menthone, (e) terpineole, (f) azone.

emi

paecter

utiAtto

saalssdR

tTsfiiaaa1

8

dbodPtpteooisttcmt

A. Goyal et al. / Computers and Ch

erformance of major drug classes via advanced computingnd physical modeling. Similar to bioinformatics, which hasmerged because of the need to comprehend the myriad of dataollected by biologists, the development of new mathematicalools, such as the one presented in this work, is paramount toxtract maximum information and benefit from in vitro drug-elease experiments.

This work also identifies a serious limitation in commonlysed fitting procedures to recover physiological model parame-ers. The optimization scheme did not converge to the true values,n all of the cases, even after reducing the input parameter space.lthough this problem may not be crucial for empirical models,

he issue is relevant for improving the physico-chemical proper-ies of a drug/formulation solely based on the relative importancef the regressed model parameters.

The use of micromodels to study the transport of drugs in thetratum corneum may complement the computational approachnd provide a more in-depth representation of the process. Suchnalysis has been applied successfully to study the fate of col-oids in porous media (Baumann & Werth, 2004). With thesemall-sized fabricated models, experimental studies of pore-cale transport processes can be visualized and help to vali-ate numerical simulation results (Corapcioglu, Chowdhury, &oosevelt, 1997).

Other in vitro techniques can be applied to estimate some ofhe model parameters before applying the regression technique.ojo, Chiang, and Chien (1987) suggested the use of intact andtripped skins to compute the partition and the diffusion coef-cients in the stratum corneum and viable skin. Their analysis

ncludes the steady-state drug penetration rate and the time lagcross the skin (Tojo et al., 1987). Binding model parameters canlso be determined from in vitro experiments and curve-fittingpproaches based on Langmuir-type equations (Hikima & Tojo,987).

. Conclusion

The parameters of a two-pathway/two-layer transdermalrug-delivery model were estimated using a regression routineased on orthogonal collocation methods. The optimal numberf discretization points, selected in the transient problem, wasetermined by solving the corresponding steady-state equations.redicted cumulative amount of drug released and concentra-

ion profiles agree well with published data. The steady-stateermeation flux through the skin was sensitive to changes inhe diffusion coefficients (DT,s, DI,s, DI,v and DT,v). “In silico”xperiments, conducted using five randomly generated valuesf these parameters, show that the orthogonal collocation-basedptimization scheme was able to recover the process values,n most cases. Transdermal penetration studies using caffeineuggest that the type of enhancer may change the relative con-ribution of possible pathways. However, as a tool to quantify

he relative contribution of each pathway, complimentary pro-edures, such as micromodels and in vitro estimation of someodel parameters, should be explored to overcome the limita-

ions of the methodology.

cal Engineering 31 (2007) 107–120 119

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